cos squared thetahttps://cos2theta.wordpress.com
Complementing the sine and squaring itMon, 16 Oct 2017 21:53:03 +0000enhourly1http://wordpress.com/https://s2.wp.com/i/buttonw-com.pngcos squared thetahttps://cos2theta.wordpress.com
Education follows enjoymenthttps://cos2theta.wordpress.com/2012/03/22/education-follows-enjoyment/
https://cos2theta.wordpress.com/2012/03/22/education-follows-enjoyment/#respondThu, 22 Mar 2012 11:18:58 +0000http://cos2theta.wordpress.com/?p=240Continue reading →]]>I had Year 7 Parent-Teacher night tonight. It’s early in the year; starting this year we have two Year 7 PT nights, which I fully support. Year 7 parents just want to know their child is settling in well and being generally happy and confident. Term 3 is too late for all that, and too late to address any problems that arise.

Anyway, several parents said their daughter really enjoys my class. I partially deflected the flattery, saying it’s because high school has dedicated Maths teachers as opposed to the generalists of primary school who love kids but don’t necessarily love mathematics. But in each case, I said my main aim with all students, but particularly Year 7, is for them to enjoy themselves in class. My motto, I cheekily declared, is “Education follows enjoyment”. I only made up the motto this evening, but there you go. Of course, I was being a little corny and showy, but the parents sincerely agreed, and usually developed the conversation along those lines.

All this would be a forgotten detail of the cut and thrust of Parent-Teacher nights, except that I reflected later how nice it is not to have parents cutting straight to the chase with the stereotypical “what does she need to do to improve her results” line of questioning. There are many stories in the Sydney media about educational issues at the moment, and you’d often be led to believe that elite private schools are full of parents thinking their money can be turned more or less directly into good results. That’s rubbish. A few are like that, and their attitude gets all the attention when it comes time for newspapers to compile their latest “Sydney’s best schools” lists. But the great majority just want their kids to be happy, to mature, to challenge themselves and be challenged, and to become a productive and contented adult. Of course they want their kids to get the best result possible, but they don’t express entitlement. And when I explicitly express my attitude that education is much broader than an exam mark, they agree wholeheartedly.

As Plato (I think) said: “There is no Roman road to Geometry”. (In modern terms: there is no shortcut to a deep education.)

And as many people have observed: when you realise how inaccurate or shallow the newspaper coverage is of an issue you are expert in, you question their coverage of other issues.

]]>https://cos2theta.wordpress.com/2012/03/22/education-follows-enjoyment/feed/0nosedogGonski reporthttps://cos2theta.wordpress.com/2012/02/20/gonski-report/
https://cos2theta.wordpress.com/2012/02/20/gonski-report/#respondMon, 20 Feb 2012 09:26:04 +0000http://cos2theta.wordpress.com/2012/02/20/gonski-report/Continue reading →]]>Apparently the following is a quote from the Gonski report into Australian government funding of school education across all sectors, released yesterday after 18 months and 7000 submissions.

Every child should have access to the best possible education, regardless of where they live, the income of their family or the school they attend.

Tell him he’s deamin’. Like anything else, money can buy a better education. Paying more doesn’t mean you necessarily get more, but it means you likely get more.

If he’s saying all schools should be equally good, then I don’t want a bar of it. There’s no way they can all be excellent. Most things in life fall into a bell curve and I doubt this will ever be any different.

It’s hard to disagree with the idea that disadvantaged schools should receive more money. But it’s important to challenge it. Why are they disadvantaged? If it’s because they have a long-running shortage of funds, then sure, more money would be great. But what if it’s because they just suck? Money alone can’t fix that.

A great school needs great leadership, a critical mass of good teachers (it’s not possible in the general case to fill the staff with “excellent” teachers), efficient administration, and money to spend on worthwhile projects. A school without good leadership, teachers or administration won’t show much in return for more money.

Of course, there are wonderful things money can do. A very interesting episode of Four Corners recently, Revolution in the classroom, showed a couple of schools that were recently quite down and out, and gave an insight into the efforts to improve them. In one (NSW) case it was a new principal with greater powers over staffing and budgets than are normally accorded public schools. In another (Vic) case, three nearby basket-case schools were razed and replaced with a fresh college. In both cases results — with the same or similar student body — are trending up. There is a focus on improving teaching within both schools. That’s not to say the teachers were bad; far from it. But in any enterprise, results will improve when you focus on improving your core processes.

So, great stories, and in one case a lot of money was required and made available. But where are the inspired and committed leaders going to come from to replicate these successes in a large number of other schools? And will the biur

So the Gonski report? Mate, if you’re going to report on school funding, leave it at that and keep the pie-in-the-sky stuff to the politicians.

I was pleased to read that independent schools will receive minimum 25% government funding using public schools as a baseline. Needier ones will get more, etc. etc. I work at a far-from-needy independent school and do not want to see my superiors grovelling for more money, or having to grovel to maintain current funding. I believe the current general rate of funding to independent schools is about right, and hopefully the Gonski recommendations will lead to a simple, transparent model so that issues can be debated on merits and not on bullshit. Key among the bullshit debating points: private schools get more federal funding than public schools. True, but irrelevant. There’s a lot of history here, but the only important thing is this: public schools get (much) more total government funding (state + federal) than independent schools, as they should. The relative amounts from different sources is completely irrelevant, and it thoroughly discredits people who push this line of argument.

The key difference is that private schools are professional organisations; public schools are industrial organisations. Both may be good or bad, but teaching is a profession and I’d rather work with professionals than industrialists.

]]>https://cos2theta.wordpress.com/2012/02/20/gonski-report/feed/0nosedogThe first topic they meet in high schoolhttps://cos2theta.wordpress.com/2012/01/14/the-first-topic-they-meet-in-high-school/
https://cos2theta.wordpress.com/2012/01/14/the-first-topic-they-meet-in-high-school/#respondSat, 14 Jan 2012 09:09:49 +0000http://cos2theta.wordpress.com/?p=68Continue reading →]]>Like most calendar years, I am teaching Year 7 in 2012. The first topic they encounter is “Whole Numbers”, which includes:

understanding the Hindu-Arabic number system, in part by comparing it to others

place value and expanded notation ()

the four operations; order of operations

divisibility tests

factors and multiples

highest common factor

prime and composite numbers

prime factors; factor trees

squares, cubes and roots

Some of those topics are quite interesting, but much of it is expected background knowledge. However, the students have come from a variety of primary schools and are now meekly having their first taste of high school mathematics. My Year 7 classes have always had a great variety in ability and background knowledge, so it becomes hard to avoid plodding through this material.

I’ve always wanted to spice it up with some fun activities, and every now and then I get an idea in my head or some inspiration from somewhere. But it’s impossible to recall all those ideas when planning an attack. Anyway, here are some ideas, in no particular order. I’d love to hear of others.

Give them a page with lots of prime numbers scattered all over it. Get them to answer questions like: write all the numbers with a digital sum of 13; write the numbers that multiply to give 731, add to give 98, etc.; write all primes between 1-10, 11-20, 21-30, etc. Think of some more, hopefully better, questions which get them immersed in the numbers and practising other skills. Next day, give them two separate quizzes: write all the prime numbers you can think of; circle all the primes in this list.

Every year we have some year-group “relays” where students answer puzzle questions in small groups. It has not escaped my notice that many of the questions practise the skills listed above. I need to track down a set of good puzzle questions.

Lots of number statement jigsaw puzzles. For example, get to the number 45 using 2, 5, 6 and 13. Answer: . Check your answer with a calculator to be sure you are obeying order of operations. Some variations: change one thing to make the answer 22 (answer: ); write and evaluate the same expression without brackets (answer: ; get as many different values as you can using the numbers 2, 5, 6 and 13 in that order.

Lots of “fill in the blank”, like . This can cover many levels of difficulty and even allow for multiple answers, like .

Learn the GCD algorithm (repeated subtraction). Not everyone will be amenable to this, so an independent worksheet for more capable students would be good. Demonstrate the algorithm a few times and get them to practise it.

Binary, octal, hexadecimal and other bases. Time permitting, you can develop a lot of understanding by really working with other bases.

This is not a bad set of ideas, but I have a frustrating feeling that I’ve been aware of many others in the past and can’t think of them now.

Pointers and ideas would be most welcome. Keep in mind that calculators are not used at this stage unless permission is given (e.g. to check answers).

]]>https://cos2theta.wordpress.com/2012/01/14/the-first-topic-they-meet-in-high-school/feed/0nosedogEnd of 2011https://cos2theta.wordpress.com/2011/12/10/end-of-2011/
https://cos2theta.wordpress.com/2011/12/10/end-of-2011/#respondFri, 09 Dec 2011 13:16:33 +0000http://cos2theta.wordpress.com/?p=65]]>I absolutely love teaching, but I also love reaching the end of the year and having a seven-week break. Boy oh boy do I deserve it! Looking forward to a new challenge in 2012: teaching the Extension course.]]>https://cos2theta.wordpress.com/2011/12/10/end-of-2011/feed/0nosedogBuilding rapporthttps://cos2theta.wordpress.com/2011/10/16/building-rapport/
https://cos2theta.wordpress.com/2011/10/16/building-rapport/#commentsSun, 16 Oct 2011 03:33:35 +0000http://cos2theta.wordpress.com/?p=58Continue reading →]]>Early this week a Year 11 student thanked me for being an awesome teacher. She came top of the class and was grateful for the support I’ve provided throughout the year. The next day another student from the same class told me I deserve a teaching award. She came near the bottom and was grateful for the support I provided in going through her exam with her after school, analysing whether she’s in the right course, and phoning her parents to discuss it with them.

So the evidence of the past two days indicates that I’m doing a good job, right? Of course, you can’t be that simplistic about it. Just as the best students are forever working out what they need to do in order to improve, to be a good teacher (or any kind of professional) you need to be constantly building your strengths and addressing your weaknesses. Plenty of students have given me grief instead of thanks or praise.

But high school teachers cop a lot of intentional and unintentional teenage crap, and student praise is a necessary corrective and a valuable encouragement. In short, and partly in jest, I’ll take whatever props I can.

Advice to beginning teachers can be quite contradictory in a lot of ways, reflecting the variety of educational contexts at play, and the fact that vastly different styles can be equally effective. One such apparent contradiction is the following two pieces of advice:

It’s not a popularity contest; it’s not important whether they like you.

It’s important to develop a good rapport with the class.

How can you develop a good rapport with them if they don’t like you? And yet these are not contradictory. Trying to be liked is problematic: it’s inappropriate and doesn’t engender respect. But if you succeed in developing a good rapport, and teach them well, then a side-effect of that will probably be that they respect you, and maybe even like you. That’s fine and healthy, and indeed precious. The affirmation in this case is earned, not sought.

Student affirmation is precious — not that I’d stoop so low as to seek it! — because it builds reputation. Teaching is, or can be, a damned difficult job, and a good reputation can make it a lot easier. A beginning teacher, or even an experienced teacher at a new school, is a blank slate. Students can be critical and the mistakes of a beginner or outsider can harm one’s reputation. First impressions last. The reputation becomes the prism through which further acts are judged. If you have a good reputation, it’s easier to get a new class on-side, build rapport, and get stuck into the business of education. Students in the class with bad attitudes will be less likely to strike out because they will sense that the majority of the class in not on their side. But if your reputation is neutral or negative, most interactions with students can be a chore.

I made such a hash of my first year it’s a wonder I’m still at the same school (soon to complete six years). My reputation nosedived and my dealings even with students unfamiliar to me could be strained. Now, after a ton of hard work over the years, I have a good reputation and the situation is entirely different. Not perfect — far from it — but much better.

On the vexed issue of how friendly to be with students, my thinking has developed over the years and is settling into an opinion. A few years ago our former principal told staff at the beginning of the year that he’d met a well-known child psychologist over the holidays. That person spoke about the properties of a good teacher, in terms of relating well to and motivating teenagers. It boiled down to “four Fs”: funny, friendly, firm and fair. It was surprising to hear funny and friendly elevated to being the properties of a good teacher. “Firm but fair” is as old as the hills, and a hard balance to strike, but funny and friendly? The principal explained: you don’t need to be funny, but approaching class time with a sense of humour and allowing students to be light-hearted is important. Same with friendliness: the teacher-student relationship is not one of friendship, but it can be friendly. Courteous, positive, interested, helpful, familiar, jocular; all that and more.

The principal’s words left a strong impression on me and gave me a framework for evaluating my rapport with students thereafter. It would be incredibly wrong to suggest that teachers must follow that framework, but it made sense to me. I think it made me feel more confident in my approach to students, and that in itself made me a more effective teacher.

So the opinion now is this: it is good to be liked as a teacher. In particular, I want students to like the experience of being in my class, because I want them to enjoy Mathematics and to carry that enjoyment beyond their school years. Not all students are naturally geared towards enjoying Mathematics themselves, but if they have a positive experience of learning it, then I hope and expect they will pass on a positive attitude to their children. I’ve seen the difference between kids whose parents like maths and kids whose parents dislike maths. In addition to working for the current generation’s education, I’m conscious of the attitudes that will influence the next generation.

I want students to look back and think “I liked Mr X”, not for my own gratification — though I’m only human — but as an indication that they received a good education.

]]>https://cos2theta.wordpress.com/2011/10/16/building-rapport/feed/2nosedogGraduation Dinner 2011https://cos2theta.wordpress.com/2011/09/21/graduation-dinner-2011/
https://cos2theta.wordpress.com/2011/09/21/graduation-dinner-2011/#respondWed, 21 Sep 2011 09:07:15 +0000http://cos2theta.wordpress.com/?p=54Continue reading →]]>Whenever I have a Year 12 class I attend the Graduation Dinner in the last week of Term 3. Well, that was last night.

One of my students has trace amounts of Aboriginal blood and consequently has been engaged in all sorts of programs that she takes with a grain of salt. She’s a boarder and I hadn’t met her parents despite having taught her for three years. So she said to me: “I want you to meet my parents. They don’t know anyone else there. I told them you’re mean and bully me because I’m Aboriginal.”

Tickled my funny bone.

Things have been slow on this blog, to be sure. There’s much to write about, but too little time. I was really pleased with the way I introduced Calculus to Year 11 this term, for instance, and will hopefully write that up soon.

]]>https://cos2theta.wordpress.com/2011/09/21/graduation-dinner-2011/feed/0nosedogFeatures of a good maths teacherhttps://cos2theta.wordpress.com/2011/04/20/features-of-a-good-maths-teacher/
https://cos2theta.wordpress.com/2011/04/20/features-of-a-good-maths-teacher/#respondTue, 19 Apr 2011 15:00:27 +0000http://cos2theta.wordpress.com/?p=48Continue reading →]]>Cathy Attard at the University of Western Sydney wrote (Oct 2010) about a longitudinal study of maths students and their perceptions of good teaching. I summarise this list for future reference. A good maths teacher, apparently:

is passionate about teaching mathematics;

responds to students’ individual needs;

gives clear explanations;

uses scaffolding rather than providing answers;

encourages positive attitudes towards mathematics; and

shows an awareness of each students’ prior knowledge.

These opinions come from 20 Year 6 students who were then followed through Years 7 and 8. There are interesting results from the study. The key messages (for me) from their high-school experience are that students’ relationship with their teacher is the most important factor in their engagement with mathematics, more so than the style of teaching; and that the overuse of technology risks depersonalising the educational experience.

The first observation comes as no great surprise to me, but it’s valuable for that message to be reinforced. (Remember it when people predict a future where teachers are replaced by robots.) The second is interesting. Using technology in the maths classroom certainly gets a range of responses from students. Even when I’m sure I’ve developed an interesting and meaningful activity, there are usually some students who don’t enjoy it and don’t see the point. These are usually students who would normally be engaged.

As for the list above, it’s interesting but I’d rather know the point of view of Year 7–12 students (preferably a separate list for each year group). Nonetheless, I find it encouraging: items 1, 3, 4 and 5 come naturally to me (well, “clear explanations” requires lots of practice…) and I don’t brush up too badly on items 2 and 6 either.

Don’t get me wrong, like any person I certainly have weak spots. It’s just nice to know that, for 20 Year 6 students out there in a different time and space, my weaknesses aren’t among the things they consider important

]]>https://cos2theta.wordpress.com/2011/04/20/features-of-a-good-maths-teacher/feed/0nosedogHung-Hsi Wu articleshttps://cos2theta.wordpress.com/2011/03/12/hung-hsi-wu-articles/
https://cos2theta.wordpress.com/2011/03/12/hung-hsi-wu-articles/#respondSat, 12 Mar 2011 02:23:46 +0000http://cos2theta.wordpress.com/?p=41Continue reading →]]>I was never all that sure about joining Twitter, but it’s paid off. My most recent post was about a very interesting article I saw mentioned on Twitter. A request to the relevant twit pointed me in the direction of heaps more by the same author. Thanks, @dcox21! I shan’t want for something to read for quite a while.

(Hung-Hsi Wu is a member of the Mathematics Department at the University of California, Berkeley. The linked articles are about high school mathematics education.)

…then they will realize how misguided it really is to teach the addition of fractions using the LCM of the denominators. p62 http://bit.ly/eLhu0y

The link is to a 70-page article by H. Wu (2008) called The Mathematics K-12 Teachers Need to Know. I’m indebted to David Cox for pointing it out, and my skimming of it reveals much to be read and thought about as time goes on. I was particularly interested in the alleged misguidedness of using the LCD (lowest common denominator) to add fractions. Seems like a great method to me! Anyway, on p. 62, Wu gives the example

and notices that the addition is carried out using the formula

Well, my senior students don’t have any great problem with fractions like that, and they use LCDs routinely to simplify algebraic fractions. What’s more, they’ve never seen a formula like the one above. Adding fractions is a first-principle process of finding a common denominator (lowest or otherwise), modifying each fraction to have that denominator, then combining. For hairy algebraic ones like

it’s a relief to find a common denominator because it makes the overall working and answer simpler.

I’m not seeing why I shouldn’t teach Year 7 students to add numeric fractions by finding a common denominator, nor teach Year 8-11 students to add algebraic fractions in the same way. It seems to me it’s a great opportunity to reinforce understanding with every example given.

One thought that occurs is that textbooks in NSW emphasise the addition of complex algebraic fractions with worthwhile LCDs, but perhaps textbooks elsewhere do not. Or perhaps Mr Wu doesn’t think such emphasis is warranted. (I could be pressed into agreeing with that…) But as it stands, for building algebraic skills and reinforcing fractional understanding, I fail to see what I’m doing wrong.

Reminder: I see much of interest in the 70 pages Wu offers, including other aspects of the teaching of fractions. I just happen to disagree with this point. At the moment.

]]>https://cos2theta.wordpress.com/2011/03/08/the-mathematics-k-12-teachers-need-to-know/feed/1nosedogChristmas pudding and ratioshttps://cos2theta.wordpress.com/2011/01/29/christmas-pudding-and-ratios/
https://cos2theta.wordpress.com/2011/01/29/christmas-pudding-and-ratios/#commentsFri, 28 Jan 2011 13:45:01 +0000http://cos2theta.wordpress.com/?p=23Continue reading →]]>I love Christmas pudding, and boiled up a nice little 100g number for dessert tonight. The instructions contain cooking times for all three sizes available. The sizes are 100g, 400g and 900g. The cooking times for oven are 20, 35 and 45 minutes respectively. The cooking times for boiling are 25, 45 and 80 minutes respectively. This is clearly not a case of linear variation, as 100:900 does not equal 20:45. Kids are likely to be at least a little surprised by that. I can imagine all sorts of activities springing from this data, depending on the year group and purported topic.

Year 7 number plane. Plot the data on a number plane (determine the scale yourself!) and join the points with a curve. Should the curve go through the origin? Predict the required cooking time for other weights.

Year 10 ratio of volumes. This is the most obvious application. Given that 100g must be boiled for 25 minutes (and given none of the other data), predict the required boiling time for the other weights and see if it matches the instructions. Required insight/knowledge: volume is proportional to mass; the different puddings are very likely to be similarly shaped; cooking time is likely proportional to the width; the width varies with the cube root of the volume (and therefore the mass). The specific language in that sentence is advances for Year 10, but the idea is simple enough.

Year 11 functions. As per Year 10 above, but plot the points, join it up and determine the equation of the curve (it “should” be , for some value of , at least to my mind). Or work out what the equation “should” be with nought but algebra, then plot it on a computer and see if it goes through all the points.

Year 8 ratios. Give the 900g cooking time and ask for a prediction for 100g. When it’s (very) wrong, give the correct answer and talk about why it might be like that. Then solicit a cooking time for 400g. Observe that cooking time is not proportional to mass, but tell them it is proportional to something to do with mass, and see if they can guess what. (Hint: it’s something on your calculator…) If possible, give an introduction to ratios of volumes of similar solids using simple diagrams, algebra and equations.

The Year 8 idea is the most handwavy, but there’s nothing wrong with that: it’s stretching the Year 8 syllabus beyond breaking point, so it depends entirely on the ability and interest of the class.